It's maybe happening already. There's plenty of games that are on GamePass but not PlayStation Now, and even more on PS Now that aren't on GamePass (though in that case that appears to be partly down to Microsoft leaning towards curation). It's entirely possible that some of the publishers/developers of those games have signed streaming exclusivity deals.
Even better, they'll all have different selections of multiplatform games, different cross-play set-ups, availability in different countries, different availability across platforms (PC/Android/iOS/proprietary streaming box/OEM streaming box/etc.), different controller compatibility, and many will offer multiple subscription tiers.
That's before we consider that we have been quite fortunate that there hasn't been any collapse of a major online gaming service (OnLive absolutely doesn't count). Services like Steam, PSN and Xbox Live are over a decade old and still preserve purchase information, friends lists, cloud saves, etc. Nintendo have rebuilt their online services a few times, sometimes quite awkwardly, and shuttered a few previously-popular storefronts, but everyone still has theoretical access to what they bought previously.
If there's a lot of competitors in the streaming market, that long run of stable services is going to vanish. One or more competitors will eventually decide it's not worth it and a lot of people are going to lose access to games, saves, and friend lists.
If game streaming really does go mainstream next generation the market is going to be a total mess.
